where \(F_{d}\)  is the hydrodynamic force in submerged conditions and \(u\)  is the particle velocity. In granular flows, the Reynolds number is usually very low, hence the hydrodynamic force can be assumed as the Stokes force \(F_{d}=3\pi \eta _{f}u d\) . During the settling process of a single particle, the particle velocity increases until the hydrodynamic force is equal to the driving force, and the particle reaches the terminal velocity \(u_{f}=Pd/(12\eta _{f})\) . In previous works [1]}, [2]}, in the inertial regime (or dry granular flows), the drag force is neglected, which assumes that particles travel with a constant acceleration \(a_{c}=3P/(2\rho _{s} d)\) , with a deduced settling time \(t_{f}=t_{dry}=\sqrt{4\rho _{s}d^2/(3P)}\)  and a time scale ratio \(t_{f}/\mathcal {T}=(2/\sqrt{3})I\) . The constant factor \(2/\sqrt{3}\approx 1.15\)  is usually ignored. In the viscous regime, we assume that the particle travels with the maximum velocity \(u_{f}=Pd/(12\eta _{f})\)  for the characteristic length \(d\) , the settling time is \(t_{f}=t_{sub}=12\eta _{f}/P\) , and the time scale ratio \(t_{f}/\mathcal {T}=12I_{v}\) . However, when the particles flow in the fluid where the inertial force is comparable to the hydrodynamic force, also known as the viscous-inertial regime, we can describe their rheology by neither the inertial number \(I\)  nor the viscous number \(I_{v}\)  individually.
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